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Trump tweeted on Monday that he was "very impressed" by the Indiana governor, who at one time had his own presidential ambitions.

"I love my job and I love my country but I'm not gonna get into hypotheticals. This was a time that we'll always cherish where we were able to spend some personal time as a family with the Trumps," Pence said Monday.

In addition to his time as governor, Pence served in the U.S. House of Representatives for 12 years.

Trump, who values loyalty, doesn't know Pence as well as some of the other candidates under consideration.

But Pence, who was once a favorite of social conservatives, has also seen his standing slide in the wake of criticism over his handling of the state's "religious freedom" law, which critics feared would allow discrimination against gays and lesbians.

After repeatedly defending it, Pence eventually signed a new version of the bill with additional protections that left both sides dissatisfied.